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Solving Compound Inequalities

Solving Compound Inequalities. Solve compound inequalities containing the word and then graph the solution. Solve compound inequalities containing the word or then graph the solution. Inequalities containing AND. Two inequalities joined by the word AND.

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Solving Compound Inequalities

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  1. Solving Compound Inequalities Solve compound inequalities containing the word and then graph the solution. Solve compound inequalities containing the word or then graph the solution.

  2. Inequalities containing AND • Two inequalities joined by the word AND. • True only if both of the inequalities are true. • The graph of a compound inequality containing the word AND is the INTERSECTION of the graphs of the two inequalities. • The solution MUST be a solution of BOTH inequalities. • The INTERSECTION can be found by determining where the graphs OVERLAP!

  3. Graph an Intersection • Graph the solution set of x < 3 and x ≥ -2. Graph first Solution -3 -2 -1 0 1 2 3 4 Graph Second Solution -3 -2 -1 0 1 2 3 4 Graph the Intersection of the two graphs -3 -2 -1 0 1 2 3 4

  4. Solve and Graph an Intersection • -5 < x – 4 < 2 • Write the inequality as an AND statement • -5 < x – 4 and x – 4 < 2 • Solve each inequality • -5 < x – 4 x – 4 < 2 • +4 +4 +4 +4 • -1 < x x < 6 • On the Next Slide, you will see the graphs.

  5. The Graphs • x > -1 (The 1st solution) • x < 6 (The 2nd solution) • -1 < x < 6 (The INTERSECTION) Use open circles because of the inequality symbols. -3 -2 -1 0 1 2 3 4 5 6 7 -3 -2 -1 0 1 2 3 4 5 6 7 This is the INTERSECTION of the two graphs! -3 -2 -1 0 1 2 3 4 5 6 7

  6. Inequalities containing OR • These compound inequalities are true if one or more of the inequalities are true. • The graph of a compound inequality containing the word OR is the UNION of the graphs of the two inequalities. • The solution of the compound inequality is a solution of either inequality. • The UNION can be found by graphing both inequalities.

  7. Write and Graph an Inequality • An airplane is experiencing turbulence while flying at 30,000 feet. The control tower tells the pilot that he should increase his altitude to at least 35,000 feet or decrease his altitude to no more than 25,000 feet to avoid the turbulence. • WORDS • The pilot needs to get the plane at least 35,000 feet or no more than 25,000 feet. • VARIABLES • Let a represent the plane’s altitude • INEQUALITY • a ≥ 35,000 or a ≤ 25,000 • Graph to follow on next Slide.

  8. GRAPH of AIRPLANE PROBLEM • a ≥ 35,000 or a ≤ 25,000 First Inequality 25,000 30,000 35,000 Second Inequality 25,000 30,000 35,000 Union 25,000 30,000 35,000

  9. Solve and Graph a Union • -3x + 4 < 19 or 7x – 3 > 18 • -4 -4 +3 +3 • -3x < 15 or 7x > 21 • -3 -3 7 7 • x > -5 or x > 3 • Graph on the Next Slide.

  10. The Graph • x > -5 or x > 3 1st Inequality -5 0 5 2nd Inequality -5 0 5 Union -5 0 5 Notice the graph of x > -5 contains all the points in x > 3. So the union is the graph of x > -5.

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